Thank you for your response.
I have just decided to book a ticket and return to the US consulate in my home country of The Netherlands to have a new H4 visa issued in my passport.
My problem was apparently that I applied for and received petition I-797A - extension of stay - while under the presumption that it was a visa-extension and being confirmed by the local USCIS office as such with the words:
"You are good to go, just staple this in your passport with your (expired) visa".
I took those words too literally and was able to leave and re-enter the country unknowingly with this form I-797A stapled in my passport 3 consecutive times, until earlier this week while entering POE San Francisco, where I received the I-193 and where it was explained to me what the problem was.

